Three of the most powerful Pokémon in Let’s Go, Pikachu! and Let’s Go, Eevee! are the Legendary Birds Articuno, Zapdos, and Moltres. Of these three, Zapdos is arguably the hardest to find, since you have to really go out of your way to find its location, whereas the other two appear in places that you’ll likely come across during your normal traveling.
To be able to get to where Zapdos is hiding, you’ll first need to have your Partner Pokémon know the Secret Technique Sea Skim, which your Partner can be taught by a man in Fuchsia City. Once your Partner has learned Sea Skim, use the Sky Dash Secret Technique (which you should also have by this point) to make your way back to Cerulean City. From here, go east towards Rock Tunnel. Just before you actually reach Rock Tunnel, you’ll notice some water just north of the Pokémon Center that’s outside the cave’s entrance. The last time you were at this area, you probably just ignored the water, since you couldn’t do anything with it. But now you have Sea Skim, so interact with the water and get surfing.
Follow this body of water all the way down until you get to a new mass of land you can land on. There will be a Poké Maniac trainer there waiting to challenge you, unless you avoid him. Go ahead and battle him for experience if you want, then make your way past him and you’ll find an abandoned power plant. Go ahead and enter. Inside, you’ll find all kinds of Electric-type Pokémon that you probably haven’t had the opportunity to catch before, so go ahead and stock up on them. The layout of the power plant is somewhat labyrinthian, but aside from a few dead ends, it’s relatively straightforward. Make your way through the power plant and at the end you’ll eventually come across the Electric/Flying-type Legendary Bird Zapdos.
Now would probably be a good time to save the game. To begin your interaction with Zapdos, walk up to it and press the A button. After a brief cutscene, you will begin your battle with Zapdos. You have to defeat Zapdos in battle before you get the chance to try to catch it, so you might want to bring along some strong Ice- or Rock-type Pokémon, as they’re super effective against the Legendary Bird. Zapdos is on Level 50 and you will only have five minutes to defeat it, so come prepared. Once you have bested it in battle, you will be given the opportunity to catch Zapdos, which works the same as catching any other wild Pokémon. Zapdos moves around quite a bit and will always have a red ring, meaning it is difficult to catch, no matter what Berries you feed it, so come prepared with tons of Ultra Balls (I would recommend at least 100) and try your best.